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Truncation

The process of truncation is a method of approximating numbers denoted as decimal expansions. It involves the deletion of all the numerals to the right of a certain point in the decimal part of an expression. Some electronic calculators use truncation to fit numbers within their displays. For example, the number 3.830175692803 can be shortened in steps as follows:

Rounding

Rounding is the preferred method of approximating numbers denoted as decimal expansions. In this process, when a given digit (call it r) is deleted at the right-hand extreme of an expression, the digit q to its left (which becomes the new r after the old r is deleted) is not changed if 0 ≤ r ≤ 4. If 5 ≤ r ≤ 9, then q is increased by 1 (''rounded up''). Most electronic calculators use rounding rather than truncation. If rounding is used, the number 3.830175692803 can be shortened in steps as follows:

Cumulative Absolute Frequency

When data are tabulated, the absolute frequencies are often shown in one or more columns. Look at Table 2-5, for example. This shows the results of the tosses of the blue die in the experiment we looked at a while ago. The first column shows the number on the die face. The second column shows the absolute frequency for each face, or the number of times each face turned up during the experiment. The third column shows the cumulative absolute frequency, which is the sum of all the absolute frequency values in table cells at or above the given position.

The cumulative absolute frequency numbers in a table always ascend (increase) as you go down the column. The total cumulative absolute frequency should be equal to the sum of all the individual absolute frequency numbers. In this instance, it is 6000, the number of times the blue die was tossed.

Cumulative Relative Frequency

Relative frequency values can be added up down the columns of a table, in exactly the same way as the absolute frequency values are added up. When this is done, the resulting values, usually expressed as percentages, show the cumulative relative frequency.

Examine Table 2-6. This is a more detailed analysis of what happened with the blue die in the above-mentioned experiment. The first, second, and fourth columns in Table 2-6 are identical with the first, second, and third columns in Table 2-5. The third column in Table 2-6 shows the percentage represented by each absolute frequency number. These percentages are obtained by dividing the number in the second column by 6000, the total number of tosses. The fifth column shows the cumulative relative frequency, which is the sum of all the relative frequency values in table cells at or above the given position.

The cumulative relative frequency percentages in a table, like the cumulative absolute frequency numbers, always ascend as you go down the column. The total cumulative relative frequency should be equal to 100%. In this sense, the cumulative relative frequency column in a table can serve as a checksum, helping to ensure that the entries have been tabulated correctly.

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